(b) Draw a free body diagram of the forces acting on a rider at the bottom
of the arc.
(c) Find the force exerted by the ride on a 60.0 kg rider and compare it to
her weight.
(d) Discuss whether the answer seems reasonable.- Unreasonable Results
A mother pushes her child on a swing so that his speed is 9.00 m/s at the
lowest point of his path. The swing is suspended 2.00 m above the child’s
center of mass.
(a) What is the centripetal acceleration of the child at the low point?
(b) What force does the child exert on the seat if his mass is 18.0 kg?
(c) What is unreasonable about these results?
(d) Which premises are unreasonable or inconsistent?
6.3 Centripetal Force
23.(a) A 22.0 kg child is riding a playground merry-go-round that is
rotating at 40.0 rev/min. What centripetal force must she exert to stay on
if she is 1.25 m from its center?
(b) What centripetal force does she need to stay on an amusement park
merry-go-round that rotates at 3.00 rev/min if she is 8.00 m from its
center?
(c) Compare each force with her weight.
24.Calculate the centripetal force on the end of a 100 m (radius) wind
turbine blade that is rotating at 0.5 rev/s. Assume the mass is 4 kg.
25.What is the ideal banking angle for a gentle turn of 1.20 km radius on
a highway with a 105 km/h speed limit (about 65 mi/h), assuming
everyone travels at the limit?
26.What is the ideal speed to take a 100 m radius curve banked at a
20.0° angle?
27.(a) What is the radius of a bobsled turn banked at 75.0° and taken at
30.0 m/s, assuming it is ideally banked?
(b) Calculate the centripetal acceleration.
(c) Does this acceleration seem large to you?
28.Part of riding a bicycle involves leaning at the correct angle when
making a turn, as seen inFigure 6.36. To be stable, the force exerted by
the ground must be on a line going through the center of gravity. The
force on the bicycle wheel can be resolved into two perpendicular
components—friction parallel to the road (this must supply the centripetal
force), and the vertical normal force (which must equal the system’s
weight).(a) Show thatθ(as defined in the figure) is related to the speedvand
radius of curvaturerof the turn in the same way as for an ideally
banked roadway—that is,θ= tan–1v^2
/
rg
(b) Calculateθfor a 12.0 m/s turn of radius 30.0 m (as in a race).
Figure 6.36A bicyclist negotiating a turn on level ground must lean at the correct
angle—the ability to do this becomes instinctive. The force of the ground on the wheel
needs to be on a line through the center of gravity. The net external force on the
system is the centripetal force. The vertical component of the force on the wheel
cancels the weight of the system while its horizontal component must supply thecentripetal force. This process produces a relationship among the angleθ, the
speedv, and the radius of curvaturerof the turn similar to that for the ideal
banking of roadways.
29.A large centrifuge, like the one shown inFigure 6.37(a), is used to
expose aspiring astronauts to accelerations similar to those experienced
in rocket launches and atmospheric reentries.(a) At what angular velocity is the centripetal acceleration 10 gif the
rider is 15.0 m from the center of rotation?
(b) The rider’s cage hangs on a pivot at the end of the arm, allowing it to
swing outward during rotation as shown inFigure 6.37(b). At what angleθbelow the horizontal will the cage hang when the centripetal
acceleration is 10 g? (Hint: The arm supplies centripetal force and
supports the weight of the cage. Draw a free body diagram of the forcesto see what the angleθshould be.)
220 CHAPTER 6 | UNIFORM CIRCULAR MOTION AND GRAVITATION
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